Special Functions in Math
In high school or college, we often learn about special functions called logarithms and exponents. As part of learning about these functions, we learned exponent laws and logarithm properties like the following:
2b + c = 2b · 2c
log(x · y) = log(x) + log(y).
For both functions, an operation between two inputs of the function resulted in an operation between the two outputs of these functions. This preservation of operations means that both exponents and logarithms are special functions called group homomorphisms.
What is Group Homomorphism?
Before jumping in and defining a group homomorphism, remember that we often represent a group using the notation (G, ∗), where G is the set and ∗ is the operation. So, for example, (R, +) is the group of real numbers under addition. Sometimes, if the operation is known or clear from context, we'll abbreviate by just calling the group G. For example, when we talk about the group of integers under addition, we usually just write Z since the integers do not form a group under any of the other basic operations.
A group homomorphism (often just called a homomorphism for short) is a function ƒ from a group (G, ∗) to a group (H, ◊) with the special property that for a and b in G,
ƒ(a ∗ b) = ƒ(a) ◊ ƒ(b)
Let's look at some examples to help make this more clear. First, let's look back at the first example. The function ƒ(x) = 2x can be considered a function from (R, +) to (R+, ·), where R+ is the set of all positive real numbers. We can tell it's a homomorphism since:
ƒ(b + c) = 2b + c = 2b · 2c = ƒ(b) · ƒ(c).
Another homomorphism that might be familiar is the map φ from Z to Z7 (the group of integers modulo 7 under addition) given by φ(x) =[x], where [x] represents x mod 7. Remember that x mod 7 is the distance of x from a multiple of 7. So, for example, 65 is 2 more than 63 (which is 7 · 9), meaning 65 mod 7 can be represented by . We can see this is a homomorphism since we know from modular arithmetic that
φ(x+y) = [x + y] = [x] + [y] = φ(x) + φ(y).
Determining a Homomorphism
A common problem when working in abstract algebra is to determine if a certain function is a homomorphism. To determine if a function is a homomorphism, we simply need to check that the function preserves the operation. In other words, we need to make sure that for a function ƒ from a group (G, ∗) to a group (H, ◊) that
ƒ(a ∗ b) = ƒ(a) ◊ ƒ(b)
is true for all a and b in G. We often can't check all of the pairs of elements in G (since it could be infinitely large). Therefore, we check to see if the above property holds for arbitrary a and b.
For example, let's determine if the function ƒ from (R, +) to (R, +) given by ƒ(x)= 5x is a homomorphism. Since there is an infinite amount of real numbers, we're going to use x and y to represent arbitrary real numbers. Let's see if we can get the homomorphism property to work.
ƒ(x + y) = 5(x + y) = 5x + 5y = ƒ(x) + ƒ(y).
It worked! Therefore, ƒ is, in fact, a homomorphism.
Let's test another function. Suppose g is a function from (R, +) to (R, +) given by g(x)=x². Let's see if it fits the homomorphism property.
g(x + y) = (x + y)² = x² + 2xy + y² = g(x) + 2xy + g(y).
Uh-oh! This means that g(x + y) = g(x) + g(y) only if 2xy = 0, something that doesn't always happen. Therefore, g is not a homomorphism.
Why are Homomorphisms Useful?
Homomorphisms are useful because they allow us to perform operations after applying a function, which means we may be able to perform a simpler calculation. Take, for example, the homomorphism φ from Z to Z7 discussed earlier. This can be used to help determine the day of the week in the future using simple calculations.
Suppose today is Tuesday, and you have a doctor's appointment in 100 days. You need to know the day of the week of the appointment so you can make sure you can get off work, because some days of the week that are easier for you to miss than others. You can get a calendar and count 100 days into the future, or you can realize that, since the days of the week repeat every 7 days, we can represent each day of the week with an integer modulo 7. Let's use 0 for Sunday, 1 for Monday, 2 for Tuesday, etc. We need to determine what 2 + 100 is mod 7. We can do this addition and then convert, or we can reduce 100 mod 7 first. Since 100 is 2 more than 98 (which is 7 · 14), 100 reduces to 2 mod 7. This gives us 2 + 2 = 4, meaning the appointment will land on a Thursday.
Let's take a few moments to review what we've learned about the special function known as homomorphism. We need to first remember that a group homomorphism is a function between two groups that maps an operation between inputs to an operation between outputs. We can determine if a function is a homomorphism by checking to see if any two elements will fit the homomorphism property. Such homomorphisms are useful since they can sometimes make difficult calculations easier, such as determining a future day of the week.
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